A regularisation method to obtain analytical solutions to the de Broglie Bohm wave equation
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| Format: | Preprint |
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2025
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| author | Kumar, Anand Aruna Srivatsa, S. K. Tengli, Rajesh |
| author_facet | Kumar, Anand Aruna Srivatsa, S. K. Tengli, Rajesh |
| contents | We develop a variational regularisation framework that enables analytical solutions of the stationary de~Broglie--Bohm wave equation. The formulation begins with a Fisher-information-augmented action functional for the probability density and phase fields, yielding the Madelung (Hamilton--Jacobi and continuity) equations and, upon complex recombination, a Schrödinger-type equation with a parametric information coupling $μ$.
Beyond this density-based formulation, we introduce a variational regularisation scheme for the de~Broglie--Bohm equations that combines a global Fisher-information regularisation at the level of the action functional with a shell-level regularisation arising from stationary flux closure. This reduction isolates the regularisation mechanism in the spatial momentum flow and yields constrained Euler--Lagrange equations governing admissible amplitude configurations. The resulting first integral possesses an elliptic structure whose admissible asymptotic branch enforces a universal canonical relation $p(x)x \to μ/2$ near amplitude zeros.
The framework yields closed-form analytical solutions for standard potentials and reveals a systematic inverse-square regularising term in the effective potential. The associated elliptic discriminant defines a geometric length scale that, for $μ=\hbar$, naturally reduces to the reduced Compton wavelength. Canonical Bohmian regularisation therefore appears as a variational admissibility condition on density dynamics, producing structurally stable analytical branches and modified yet consistent energy spectra within stationary dBB mechanics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_18555 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A regularisation method to obtain analytical solutions to the de Broglie Bohm wave equation Kumar, Anand Aruna Srivatsa, S. K. Tengli, Rajesh Quantum Physics We develop a variational regularisation framework that enables analytical solutions of the stationary de~Broglie--Bohm wave equation. The formulation begins with a Fisher-information-augmented action functional for the probability density and phase fields, yielding the Madelung (Hamilton--Jacobi and continuity) equations and, upon complex recombination, a Schrödinger-type equation with a parametric information coupling $μ$. Beyond this density-based formulation, we introduce a variational regularisation scheme for the de~Broglie--Bohm equations that combines a global Fisher-information regularisation at the level of the action functional with a shell-level regularisation arising from stationary flux closure. This reduction isolates the regularisation mechanism in the spatial momentum flow and yields constrained Euler--Lagrange equations governing admissible amplitude configurations. The resulting first integral possesses an elliptic structure whose admissible asymptotic branch enforces a universal canonical relation $p(x)x \to μ/2$ near amplitude zeros. The framework yields closed-form analytical solutions for standard potentials and reveals a systematic inverse-square regularising term in the effective potential. The associated elliptic discriminant defines a geometric length scale that, for $μ=\hbar$, naturally reduces to the reduced Compton wavelength. Canonical Bohmian regularisation therefore appears as a variational admissibility condition on density dynamics, producing structurally stable analytical branches and modified yet consistent energy spectra within stationary dBB mechanics. |
| title | A regularisation method to obtain analytical solutions to the de Broglie Bohm wave equation |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2512.18555 |